The bill could be considered as early as next week, depending on when Reid calls it to the floor, the aide said.

“The American people have spoken very loudly and very clearly. They have told the Congress to restore those cuts to military retirees and we have listened,” Sanders, chairman of the Senate Veterans Committee, said on Thursday.

Congressional aides say restoring the cuts isn't a done deal, as the issue of how to offset the cuts must still be resolved. But it is expected that Reid and Sanders will find offsets within the Department of Veterans Affairs' (VA) budget.

“It’s just a matter of working out the details on that,” the Senate aide said.

Possible offsets under the Sanders proposal would “not cut other veterans programs” within the VA's budget, and instead would come from inefficiencies and by shifting money around, said a Sanders aide.

The VA was appropriated $63.2 billion in the recent 2014 omnibus spending bill.

Still, an offset would be approximately a tenth of the budget at a department that is focused on its own challenges.

The VA is currently developing an electronic health records system with the Department of Defense that would allow for seamless transfer of medical information between the two departments.

And it has an estimated 6.5 million patients to treat in 2014, and has set an ambitious goal to reduce a backlog of veterans’ claims by 2015.

However, veterans groups say there is enough money in the VA budget for offsets.

“It hasn’t been a funding problem for years – it’s an execution problem,” said Pete Hegseth, CEO of Concerned Veterans for America.

Veterans service organizations and advocates have mounted fierce opposition to the cuts, which are slated to begin December 2015 and hold the cost-of-living adjustment in pensions to 1 percent below inflation until age 62.

Restoring the cuts has widespread support in Congress: about one-in-three members support it.

More than a dozen members of Congress have offered bills to restore the cuts, but so far, none have found an offset that is acceptable to both Republicans and Democrats.

A proposal by Rep. Jeff Miller (R-Fla.), chairman of the House Veterans Affairs Committee, would reverse the cuts without an offset, which some lawmakers say is a non-starter.

If the veterans bill moves through the Senate, it would also have to pass the House to get to President Obama’s desk.

The spending bill passed last week had reversed the cuts for medically-retired veterans of working age, but not for others.

Rep. Richard Nugent (R-Fl.) accused other members of Congress for crying “crocodile tears” over the COLA cuts – vowing to reverse them, but voting for an appropriations bill that kept them in place.

“As a nation, we don’t break promises that have already been made to our troops. It’s not just about the dollars, it’s about the principle,” said Nugent in a statement on his website.

“It’s not like anyone’s getting rich off these pensions,” said Hegseth, who added that there are better ways to reform military benefits and compensation. “We could be OK with a deal with shared sacrifices.

“It's that no one's done anything about the deficit for years, and then military retirees are the first ones on the chopping block,” he said.

The two-year budget deal also includes $6 billion in cuts to newly-hired federal worker pensions, as well as other savings. But it reduces planned automatic spending cuts known as the sequester by about $60 billion over two years, and increases the overall budget compared to what it would have been with the sequester.

The pension cuts are demanding attention from top Pentagon leadership, who will testify next Tuesday at a Senate Armed Service Committee hearing.

Pentagon leaders have also appointed a congressionally mandated commission that will look at changes to military benefits and issue recommendations by February 2015.